发表机构
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明最小匹配覆盖非二部图至多有5(|V(G)|-2)/2条边，且该界对所有|V(G)|≥6紧，推广了二部图情形。
AI 中文摘要
一个具有至少两个顶点的连通图 $G$ 如果其每条边都位于某个完美匹配中，则称为匹配覆盖图。如果删除任意一条边后得到的图不再是匹配覆盖图，则该匹配覆盖图是最小的。Lovász 和 Plummer [J. Combin. Theory, Ser. B 23 (1977) 127--138] 通过耳分解证明了：每个不同于 $K_2$ 的最小匹配覆盖二部图 $G$ 至多有 $(3|V(G)|-6)/2$ 条边，并且该界对所有 $|V(G)|\ge4$ 都是紧的。在本文中，我们证明：每个具有至少 6 个顶点的最小匹配覆盖非二部图 $G$ 至多有 $5(|V(G)|-2)/2$ 条边，并且该界对所有 $|V(G)|\ge6$ 都是紧的。
英文摘要
A connected graph $G$ with at least two vertices is {\em matching covered} if each of its edges lies in a perfect matching. A matching covered graph is {\em minimal} if the removal of any edge results in a graph that is no longer matching covered. Lovász and Plummer [J. Combin. Theory, Ser. B 23 (1977) 127--138] proved by ear decompositions that every minimal matching covered bipartite graph $G$ different from $K_2$ has at most $(3|V(G)|-6)/2$ edges, and this bound is sharp for all $|V(G)|\ge4$. In this paper, we prove that every minimal matching covered nonbipartite graph $G$ with at least 6 vertices has at most $5(|V(G)|-2)/2$ edges, and this bound is sharp for all $|V(G)|\ge6$.
